Affiliations: [a] Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, A 8010 Graz, Austria. E-mail: holzmann@math.tugraz.at | [b] Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 250 68, Řež near Prague, Czechia. E-mail: lotoreichik@ujf.cas.cz
Abstract: In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in L2(R2). The proofs rely on perturbation theory of linear operators, Floquet–Bloch analysis, and properties of Schrödinger operators with point interactions.
